The function \(\rho (\beta ;\sigma )\)  was first derived in [1]} to describe the detection boundary of sparse heteroscedastic normal mixtures; see the discussion in Section  below. Figure REF , adapted from [1]}, depicts \(\rho (\beta ;\sigma )\)  for several choices of \(\sigma \) . Theorem REF  extends the results of [1]} to the family of APLC models, several of which we discuss in Section . The function \(\rho (\beta ;\sigma =1)\)  was used in [4]}, [5]}, [6]}, [7]}, [8]} to characterizes the detection boundary of several rare/weak settings.
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